When Curtis Arnold launched the card-comparison website BestPrepaidDebitCards.com in early 2013, American Express Co. signed on as an early advertiser.

Soon after, Mr. Arnold said, the company asked him to refer to its widely used Bluebird card not as a prepaid card but as a "checking and debit alternative." Mr. Arnold agreed. Then, last summer, American Express executives urged the site to change how it presented the...